In the following exercises, translate to a system of...

In the following exercises, translate to a system of equations and solve the system.
A married couple together earn $\$ 75,000$. The husband earns $\$ 15,000$ more than five times what his wife earns. What does the wife earn?

Key Concepts

Variables

Variables are symbols used to represent unknown quantities in mathematical expressions and equations. They allow generalization of problems and facilitate the process of setting up equations based on word problem descriptions.

Linear Equations

Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable. They describe relationships with constant rates of change and are typically solved using methods such as substitution or elimination.

Substitution Method

The substitution method is a systematic way to solve a system of equations by expressing one variable in terms of another from one equation and then substituting that expression into the other equation(s). This method simplifies the system and is particularly useful when one equation is already solved for one of the variables.

Translation of Word Problems

This concept involves converting a word problem's description into mathematical equations by identifying the unknown quantities and expressing the relationships between them using variables. It is essential for framing real-world situations into a structure that can be mathematically analyzed and solved.

System of Equations

A system of equations is a set of two or more equations with multiple variables that are treated simultaneously. The goal is to find values for the variables that satisfy all equations in the system, making it a fundamental tool for resolving problems with multiple interdependent conditions.

Transcript

00:02 Okay, so we're going to be solving for systems of equations today, but instead of being explicitly given two equations to solve for, we're given a word problem, which means we first have to read the problem, understand the question, identify the variables, write the equations, and then solve the equations.

00:18 So let's start.

00:19 So here we are given a married couple, a wife and a husband.

00:24 So those are variables.

00:25 So let's name those two variables.

00:28 So the variables are husband and wife.

00:34 Let's assign the husband to be x and the wife to be y.

00:40 So it says they together earn $75 ,000, which means our first equation is going to be x plus y equals $75 ,000.

00:52 All right.

00:54 The husband earns 15 ,000 more than five times his wife earns.

00:58 Okay.

00:58 The husband is x.

01:00 That means x is going to be 15 ,000 more than five times what the wife earns.